An Optimal Algorithm for Plane Matchings in Multipartite Geometric Graphs

  • Ahmad Biniaz
  • Anil Maheshwari
  • Subhas C. Nandy
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

Let P be a set of n points in general position in the plane which is partitioned into color classes. P is said to be color-balanced if the number of points of each color is at most \(\lfloor n/2\rfloor \). Given a color-balanced point set P, a balanced cut is a line which partitions P into two color-balanced point sets, each of size at most \(2n/3 + 1\). A colored matching of P is a perfect matching in which every edge connects two points of distinct colors by a straight line segment. A plane colored matching is a colored matching which is non-crossing. In this paper, we present an algorithm which computes a balanced cut for P in linear time. Consequently, we present an algorithm which computes a plane colored matching of P optimally in \(\Theta (n\log n)\) time.

Keywords

Convex Hull General Position Perfect Match Blue Point Colored Match 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    The 1979 Putnam exam. In: Alexanderson, G.L., Klosinski, L.F., Larson, L.C. (eds.) The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965–1984. Mathematical Association of America, USA (1985)Google Scholar
  2. 2.
    Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29(3), 912–953 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aichholzer, O., Cabello, S., Monroy, R.F., Flores-Peñaloza, D., Hackl, T., Huemer, C., Hurtado, F., Wood, D.R.: Edge-removal and non-crossing configurations in geometric graphs. Disc. Math. & Theo. Comp. Sci. 12(1), 75–86 (2010)MATHGoogle Scholar
  4. 4.
    Bereg, S., Hurtado, F., Kano, M., Korman, M., Lara, D., Seara, C., Silveira, R.I., Urrutia, J., Verbeek, K.: Balanced partitions of 3-colored geometric sets in the plane. Discrete Applied Mathematics 181, 21–32 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bereg, S., Kano, M.: Balanced line for a 3-colored point set in the plane. Electr. J. Comb. 19(1), P33 (2012)MathSciNetGoogle Scholar
  6. 6.
    Cherkassky, B.V., Goldberg, A.V., Silverstein, C.: Buckets, heaps, lists, and monotone priority queues. SIAM J. Comput. 28(4), 1326–1346 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443 (1990)Google Scholar
  8. 8.
    Hershberger, J., Suri, S.: Applications of a semi-dynamic convex hull algorithm. BIT 32(2), 249–267 (1992)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kano, M., Suzuki, K., Uno, M.: Properly colored geometric matchings and 3-trees without crossings on multicolored points in the plane. In: Akiyama, J., Ito, H., Sakai, T. (eds.) JCDCGG 2013. LNCS, vol. 8845, pp. 96–111. Springer, Heidelberg (2014) Google Scholar
  10. 10.
    Lo, C., Matousek, J., Steiger, W.L.: Algorithms for ham-sandwich cuts. Discrete & Computational Geometry 11, 433–452 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Overmars, M.H., van Leeuwen, J.: Maintenance of configurations in the plane. J. Comput. Syst. Sci. 23(2), 166–204 (1981)CrossRefMATHGoogle Scholar
  12. 12.
    Sitton, D.: Maximum matchings in complete multipartite graphs. Furman University Electronic Journal of Undergraduate Mathematics 2, 6–16 (1996)Google Scholar
  13. 13.
    Vaidya, P.M.: Geometry helps in matching. SIAM J. Comput. 18(6) (1989)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ahmad Biniaz
    • 1
  • Anil Maheshwari
    • 1
  • Subhas C. Nandy
    • 2
  • Michiel Smid
    • 1
  1. 1.Carleton UniversityOttawaCanada
  2. 2.Indian Statistical InstituteKolkataIndia

Personalised recommendations